Non-Adaptive Adaptive Sampling on Turnstile Streams
Adaptive sampling is a useful algorithmic tool for data summarization problems in the classical centralized setting, where the entire dataset is available to the single processor performing the computation. Adaptive sampling repeatedly selects rows of an underlying matrix $\mathbf{A}\in\mathbb{R}^{n...
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Zusammenfassung: | Adaptive sampling is a useful algorithmic tool for data summarization
problems in the classical centralized setting, where the entire dataset is
available to the single processor performing the computation. Adaptive sampling
repeatedly selects rows of an underlying matrix
$\mathbf{A}\in\mathbb{R}^{n\times d}$, where $n\gg d$, with probabilities
proportional to their distances to the subspace of the previously selected
rows. Intuitively, adaptive sampling seems to be limited to trivial multi-pass
algorithms in the streaming model of computation due to its inherently
sequential nature of assigning sampling probabilities to each row only after
the previous iteration is completed. Surprisingly, we show this is not the case
by giving the first one-pass algorithms for adaptive sampling on turnstile
streams and using space $\text{poly}(d,k,\log n)$, where $k$ is the number of
adaptive sampling rounds to be performed.
Our adaptive sampling procedure has a number of applications to various data
summarization problems that either improve state-of-the-art or have only been
previously studied in the more relaxed row-arrival model. We give the first
relative-error algorithms for column subset selection, subspace approximation,
projective clustering, and volume maximization on turnstile streams that use
space sublinear in $n$. We complement our volume maximization algorithmic
results with lower bounds that are tight up to lower order terms, even for
multi-pass algorithms. By a similar construction, we also obtain lower bounds
for volume maximization in the row-arrival model, which we match with
competitive upper bounds.
See paper for full abstract. |
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DOI: | 10.48550/arxiv.2004.10969 |